Quasi-derivations of the (extended) Schrödinger--Witt algebras

The Schrödinger–Witt algebras are infinite-dimensional Lie algebras of local dynamical symmetries of the free Schrödinger equation; they contain the Witt algebra of conformal transformations of the time line and appear in non-equilibrium statistical physics and in non-relativistic conformal field theory. We describe all quasi-derivations of the original and the twisted Schrödinger–Witt algebras. We prove that every quasi-derivation is the sum of a derivation, a trivial $\frac{1}{2}$-derivation, a linear map with values in the center, and a finite linear combination of maps from three new families of quasi-derivations. As corollaries, we describe all $\delta$-derivations, all $\delta$-biderivations and all transposed $\delta$-Poisson structures of the Schrödinger–Witt algebras. Finally, we prove that every quasi-derivation of the (original or twisted) extended Schrödinger–Witt algebra is the sum of an inner derivation and a multiple of the identity map. As a consequence, there are no nontrivial $\delta$-derivations (for $\delta \mathrm{\notin }\{\frac{1}{2},1\}$), $\delta$-biderivations (for $\delta \mathrm{\ne }1$), and transposed $\delta$-Poisson structures on these algebras.